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ShiftandInvert.
None of the transformations to standard form, (8.3),
(8.4), or (8.5), can be used when both
and are singular or when is illconditioned.
An attractive and popular technique is to apply first the shift to the
original problem and then carry out the splitinvert. This is the
SI, as discussed
in §3.3. Specifically, let be a userselected
shift such that the matrix is nonsingular; then the
original problem (8.1) can be transformed to

(219) 
where
We see that the eigenvalues of the
problem (8.1) closest to the shift are mapped
as the exterior eigenvalues of the reduced standard eigenvalue
problem (8.6), that is, to the eigenvalues of largest magnitude,
and these are the eigenvalues that are first well approximated by the
iterative methods.
In practice, an effective shift selection depends on the user's
preferences and on knowledge
of the underlying generalized eigenproblem.
A good shift not only amplifies the
desired eigenvalues, but it also leads to a wellconditioned
matrix . This often makes the task of selecting good shifts
a challenging one.
For the application of an iterative method for the reduced standard
eigenvalue problem (8.6), one needs to evaluate
matrixvector products
or
for given vectors and .
For an efficient evaluation, let

(220) 
represent some convenient factorization of , where and
are square matrices. Since is assumed
to be nonsingular, the factors and are
also nonsingular. The factorization should be chosen so that
the corresponding
linear systems of equations with ,
and/or ,
can be solved
efficiently, and typically, sparse LU factorizations are used.
See §10.3.
Of course, one can also select
and
if this leads to convenient linear systems.
With the above factorization, the matrixvector product
can be evaluated as follows:
(a) form ,
(b) solve for ,
(c) solve for .
Similar, the matrixvector product
can be evaluated as
following three steps:
(a) solve for ,
(b) solve for ,
(c) form .
The SI technique is a powerful tool
in the treatment of the generalized eigenvalue
problem (8.1). The major problem, which often becomes bottleneck,
is to find a convenient factorization (8.7) of
so that the associated linear systems of equations
can be solved efficiently. If accurate solution of the linear systems with
becomes too expensive, then one may consider the usage of
inexact Cayley transformations (see §11.2), or the
JacobiDavidson method.
Next: JacobiDavidson Method G. Sleijpen and
Up: Transformation to Standard Problems
Previous: Splitandinvert .
Contents
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Susan Blackford
20001120